A Variational Perspective on Accelerated Methods in Optimization (1603.04245v1)

Abstract: Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. While many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the \emph{Bregman Lagrangian} which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods correspond to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.

• The paper presents the Bregman Lagrangian as a unifying framework that bridges continuous-time dynamics and discrete accelerated methods.
• It employs the Euler-Lagrange equations to derive second-order differential equations that achieve polynomial and exponential convergence rates.
• The authors highlight the time-dilation property of the Bregman Lagrangian, inspiring new strategies for discretizing and accelerating optimization algorithms.

A Variational Perspective on Accelerated Methods in Optimization

The paper "A Variational Perspective on Accelerated Methods in Optimization" by Andre Wibisono, Ashia C. Wilson, and Michael I. Jordan presents a comprehensive paper of accelerated optimization methods through the lens of a continuous-time framework. It introduces the Bregman Lagrangian, a functional that serves to generate a broad class of accelerated methods, including widely used algorithms like accelerated gradient descent and its non-Euclidean and higher-order extensions.

Core Contributions

The main thrust of the paper is the establishment of a continuous-time perspective, which provides new insights into the systematic derivation of accelerated methods. Previous approaches to deriving such methods primarily relied on intuitive or case-specific algebraic manipulations. The authors leverage a variational approach, observing that these methods correspond to solutions that travel along a specific curve in space-time at different speeds.

1. Bregman Lagrangian: The introduction of the Bregman Lagrangian, which encapsulates a large family of accelerated methods, is central to the paper's contribution. This functional bridges the gap between continuous-time curves and discrete-time algorithms.
2. Euler-Lagrange Framework: By employing a continuous-time variational approach, the authors derive second-order differential equations whose solutions correspond to accelerated optimization paths in continuous-time space.
3. Time-Dilation Property: A notable theoretical insight is that the Bregman Lagrangian maintains its form under time dilation, meaning one can transform curves to travel at different speeds, linking various accelerated methods.

Numerical Analysis and Theoretical Implications

The paper provides a rigorous analysis of the convergence rates associated with specific choices of the Lagrangian parameters:

• Polynomial family: For this class, they demonstrate the corresponding Euler-Lagrange equations achieve $O(1/t^p)$ convergence rates. The intricate process of discretizing these continuous-time dynamics to create algorithms with matching $O(1/k^p)$ rates is explored in depth.
• Exponential family: The authors also discuss a subfamily with exponential convergence rates, $O(e^{-ct})$, although finding discrete equivalents proved to be less straightforward compared to the polynomial case.

Theoretical and Practical Implications

The variational framework not only enhances theoretical understandings but also has practical implications for designing new optimization algorithms. The ability to view and derive accelerated methods from a continuous-time perspective allows for potentially crafting more efficient discretization techniques in the future.

The Bregman-Lagrangian framework provides a systematic method to relate discrete algorithms and continuous dynamics, potentially leading to novel constructions in various optimization settings beyond those explicitly covered. The paper’s insights into the time-dilation properties suggest possibilities for feature extraction from canonical accelerated paths, which can be especially advantageous in optimization tasks that involve composite, stochastic, or nonconvex structures.

Future Directions

The framework proposed opens avenues for future research, particularly in extending the acceleration techniques to other areas such as stochastic optimization and exploring connections to Hamiltonian dynamics. Additionally, potential exists for improving the understanding of the transition from continuous-time to discrete-time dynamics, which could bridge more exotic settings or functions.

In conclusion, this paper provides a rigorous mathematical framework that enhances the understanding of accelerated optimization methods using a variational approach. Its contributions are theoretically significant, offering a new way to conceptualize and derive these increasingly critical methods in optimization theory.